Motivation Outline Background Our Calculational Method Cross-Checks On The Result The Structure Of The Small x Limit Outlook The Two-Loop Soft Function For Fully Differential Continuum Top Quark Pair Production At Future Linear Colliders Robert M. Schabinger with Andreas von Manteuffel and Hua Xing Zhu The PRISMA Cluster of Excellence and Mainz Institute of Theoretical Physics Two-Loop Fully Differential Continuum e + e − → t ¯ Robert M. Schabinger t
Motivation Outline Background Our Calculational Method Cross-Checks On The Result The Structure Of The Small x Limit Outlook Motivation A future high-energy linear collider such as the proposed International Linear Collider (ILC) will provide an ideal environment for precision top quark physics. An ILC center-of-mass energy of 500 GeV is often discussed, for example in the context of Zt ¯ t form factor measurements. In fact, there has even been a proposal to measure the top Yukawa ( Ht ¯ t ) coupling at a center-of-mass energy of 1 TeV ( Roloff and Strube, LCD-NOTE-2013-001 ) which requires, among other things, precise control over the t ¯ t background. Two-Loop Fully Differential Continuum e + e − → t ¯ Robert M. Schabinger t
Motivation Outline Background Our Calculational Method Cross-Checks On The Result The Structure Of The Small x Limit Outlook Motivation A future high-energy linear collider such as the proposed International Linear Collider (ILC) will provide an ideal environment for precision top quark physics. An ILC center-of-mass energy of 500 GeV is often discussed, for example in the context of Zt ¯ t form factor measurements. In fact, there has even been a proposal to measure the top Yukawa ( Ht ¯ t ) coupling at a center-of-mass energy of 1 TeV ( Roloff and Strube, LCD-NOTE-2013-001 ) which requires, among other things, precise control over the t ¯ t background. At these energies, continuum t ¯ t production is important and cannot be safely ignored! Two-Loop Fully Differential Continuum e + e − → t ¯ Robert M. Schabinger t
Motivation Outline Background Our Calculational Method Cross-Checks On The Result The Structure Of The Small x Limit Outlook Outline 1 Motivation 2 Background The Factorization Formula What We Have Calculated 3 Our Calculational Method Get The Squared Amplitude From Feynman Diagrams Apply Integration By Parts Reduction To The Integrand Integrate The Masters Using Henn Auxiliary Systems Derive All-Orders-in- ǫ Expressions For Input Integrals 4 Cross-Checks On The Result 5 The Structure Of The Small x Limit 6 Outlook Two-Loop Fully Differential Continuum e + e − → t ¯ Robert M. Schabinger t
Motivation Outline Background The Factorization Formula Our Calculational Method What We Have Calculated Cross-Checks On The Result The Structure Of The Small x Limit Outlook Factorization in the Threshold Region Eichten and Hill, Phys. Lett. B234 , 511 (1990); Grinstein, Nucl. Phys. B339 , 253 (1990); Isgur and Wise, Phys. Lett. B237 , 527 (1990); Georgi, Phys. Lett. B240 , 447 (1990) In the threshold region where the energy of the QCD radiation off of the top quarks is small, heavy quark effective theory (HQET) implies that t ¯ t differential distributions factorize, e.g. � � m t �� � � 2 E cut �� dσ t ¯ dσ t ¯ t t d cos θH t ¯ Σ t ¯ 0 t t d cos θ = x, ln x, ln + O ( E cut /m t ) µ µ � 1 − 4 m 2 1 − t s In the above , x = � 1 − 4 m 2 1 + t s Two-Loop Fully Differential Continuum e + e − → t ¯ Robert M. Schabinger t
Motivation Outline Background The Factorization Formula Our Calculational Method What We Have Calculated Cross-Checks On The Result The Structure Of The Small x Limit Outlook The Two-Loop Soft Function � � 2 E cut �� � E cut t ( x, λ, µ ) Σ t ¯ dλ S t ¯ t x, ln = µ 0 � � � t ( x, λ, µ ) = 1 S t ¯ n | X s �� X s | Y † n Y † δ λ − E Xs � 0 | Y n Y ¯ n | 0 � ¯ N c Xs n 2 = 4 m 2 n = 2 − 4 m 2 n 2 = ¯ t t n · ¯ s s Note that the hard function is known to two-loop order ( Bernreuther et. al. , Nucl. Phys. B706 , 245 (2005), Nucl. Phys. B712 , 229 (2005), and Nucl. Phys. B723 , 91 (2005); Gluza et. al. JHEP 0907 , 001 (2009) ) but an appropriate two-loop, fully differential, full QCD program is not yet available. Two-Loop Fully Differential Continuum e + e − → t ¯ Robert M. Schabinger t
Motivation Outline Get The Squared Amplitude From Feynman Diagrams Background Apply Integration By Parts Reduction To The Integrand Our Calculational Method Integrate The Masters Using Henn Auxiliary Systems Cross-Checks On The Result Derive All-Orders-in- ǫ Expressions For Input Integrals The Structure Of The Small x Limit Outlook (Carefully) Evaluate The Appropriate Squared Sum of Cut Eikonal Feynman Diagrams + · · · + + · · · + Two-Loop Fully Differential Continuum e + e − → t ¯ Robert M. Schabinger t
Motivation Outline Get The Squared Amplitude From Feynman Diagrams Background Apply Integration By Parts Reduction To The Integrand Our Calculational Method Integrate The Masters Using Henn Auxiliary Systems Cross-Checks On The Result Derive All-Orders-in- ǫ Expressions For Input Integrals The Structure Of The Small x Limit Outlook Integration By Parts Reduction Tkachov, Phys. Lett. B100 , 65, (1981); Chetyrkin and Tkachov, Nucl. Phys. B192 , 159, (1981) It is well-known that one can generate recurrence relations by considering families of Feynman integrals and then integrating by parts in d spacetime dimensions, e.g. � � � d d ℓ ∂ ℓ µ 0 = ( ℓ 2 − m 2 ) a (2 π ) d ∂ℓ µ � � � d d ℓ 2 aℓ 2 d = ( ℓ 2 − m 2 ) a − ( ℓ 2 − m 2 ) a +1 (2 π ) d ( d − 2 a ) I ( a ) − 2 am 2 I ( a + 1) = In this case, the recurrence relation can be solved explicitly but it is one of the few known examples where one can proceed directly. Two-Loop Fully Differential Continuum e + e − → t ¯ Robert M. Schabinger t
Motivation Outline Get The Squared Amplitude From Feynman Diagrams Background Apply Integration By Parts Reduction To The Integrand Our Calculational Method Integrate The Masters Using Henn Auxiliary Systems Cross-Checks On The Result Derive All-Orders-in- ǫ Expressions For Input Integrals The Structure Of The Small x Limit Outlook Apply the Reduze 2 Integration By Parts Identity Solver To Reduce The Integrand In all but the simplest examples, the strategy used ( Laporta, Int. J. Mod. Phys. A15 , 5087, (2000) ) to solve integration by parts identities is to build a linear system of equations for the Feynman integrals in the calculation by explicitly substituting particular values of the indices into the recurrence relations. The Reduze 2 ( von Manteuffel and Studerus, arXiv:1201.4330 ) implementation of Laporta’s algorithm is robust and well-tested. However, the public version of the code was written with virtual corrections in mind and does not support phase space integrals such as those which arise in the calculation under discussion. Two-Loop Fully Differential Continuum e + e − → t ¯ Robert M. Schabinger t
Motivation Outline Get The Squared Amplitude From Feynman Diagrams Background Apply Integration By Parts Reduction To The Integrand Our Calculational Method Integrate The Masters Using Henn Auxiliary Systems Cross-Checks On The Result Derive All-Orders-in- ǫ Expressions For Input Integrals The Structure Of The Small x Limit Outlook Apply the Reduze 2 Integration By Parts Identity Solver To Reduce The Integrand In all but the simplest examples, the strategy used ( Laporta, Int. J. Mod. Phys. A15 , 5087, (2000) ) to solve integration by parts identities is to build a linear system of equations for the Feynman integrals in the calculation by explicitly substituting particular values of the indices into the recurrence relations. The Reduze 2 ( von Manteuffel and Studerus, arXiv:1201.4330 ) implementation of Laporta’s algorithm is robust and well-tested. However, the public version of the code was written with virtual corrections in mind and does not support phase space integrals such as those which arise in the calculation under discussion. The functionality of the code is straightforward to appropriately extend and we find that there are just 14 master integrals which need to be calculated! Two-Loop Fully Differential Continuum e + e − → t ¯ Robert M. Schabinger t
Recommend
More recommend
